M-matrix

Remarks

Every M-matrix is a Z-matrix. For a Z-matrix the choice of $s$ does not matter: whenever $s \ge \max_i a_{ii}$, the number $s - \rho(sI - A)$ is the same. Example: $A = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} = 2I - B$ with $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. The eigenvalues of $B$ are $\pm 1$, so $\rho(B) = 1 < 2$, and $A$ is a nonsingular M-matrix. Non-example: $A = \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix} = I - B$ with $B = \begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}$. Here $\rho(B) = 2 > 1$, so clause 3 fails; indeed $A$ has the negative eigenvalue $-1$. For a Z-matrix $A$, each of the following conditions is equivalent to $A$ being a nonsingular M-matrix.

1. $A$ is invertible and $A^{-1}$ is nonnegative.
2. Every eigenvalue of $A$ has positive real part.
3. All leading principal minors of $A$ are positive.